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d#
WSRC-MS-98-00246
&)F--q g 67 --Structural Integrity Justification of a Rectangular Duct Under
Large External Pressure
byG. E. Driesen
Westinghouse Savannah River Company
Savannah River Site
Aiken, South Carolina 29808
D. A. Amin
T. T. Wu
A document prepared for 1998AMERICAN SOCIETY OF MECHANICAL ENGINEERS, PRESSURE VESSEL
AND PIPING CONFERENCE at San Diego, CA, USA from 7/26/98 - 7/30/98.
DO€ ContractNo. DE-AC09-96SR18500
This paper was prepared n connection with work don0 under the above contract number with the U.S.
Department of Energy. By acceptance of this paper, the publisher and/or recipient acknowledges the U. S.
Government's right to retain a nonexclusive, royalty-free icense in and to any copyright covering this paper,
along with the right to reproduce and to authorize others to reproduce all or part of the copyrighted paper.
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c
STRUCTURAL INTEGRITY JUSTIFICATION OF A RECTANGULAR DUCTUNDER LARGE EXTERNAL PRESSURE
Tsu-te Wu
Westinghouse Savannah River Company
Aiken, South Carolina 29808
Gary E. Driesen
Westinghouse Savannah River Company
Aiken, South Carolina 29808
Dilip A. Amin
Westinghouse Savannah River Company
Aiken, South Carolina 29808
ABSTRACT
This paper presents a methodology to justify the structural
integrity of a 116”x40” rectangular HVAC duct subjected to
tornado-induced depressurization. Hand calculations are unable to
demonstrate structural qualification per ASME AG-1 criteria.
Based on these calculations, a significant number of additional
reinforcing stiffeners would be required. As an alternative to this
design gpgrade, a finte element analysis (FEA) is performed to
eliminate excess analysis conservatism by considering both material
and geometrical nonlinearity.
The load-deflection relationship is determined using he elastic-
plastic and large-deflection analysis capabilities of the ABAQUS
computer code. The allowable collapse load based on the ASME
Code, Section ID, Appendix F for Level-D Service is much greater
than the FEA computed collapse load. Co dma tion of the HVAC
duct structural adequacy of the duct is therefore possible without a
design modification.
For comparative purposes, linear elastic and elastic-plastic,
small-deflection FEA evaluations are also performed. A comparison
of the results shows that the effects of large deflections are
important considerations in evaluating the structural capability of
W A C ducts under large pressures.
1.0 INTRODUCTION
The consequences of tomado-induced depressurization on a
large rectangular HVAC duct section represents a difficult problem
to evaluate. Closed form solutions of duct panel sections based ontraditioiial elastic plate and beam theory are often too conservative
to enabte verification of structural adequacy. To enable tractable
analyses, simplifying assumptions are typically made which neglect
the effects of membrane stihess, structural stability, and stre
redistribution.
Structural design criteria for WA C duct are currently based
national Codes and Standards developed by the SMAcNA1s2
The ASME AG-1 design code considers dynamic pressu
loads (DPD) resulting ftom a design basis accident, such as
tornado, to be classified as ServiceLevel-D. AG-1 further stipula
(AA-4332.3) that Level-D design verification of linear-type syste
is ensured through compliance with ASME Code, SectionID.
This paper presents a methodology to verify the structu
integrity of an existing 116”x 40” rectangular HVAC duct w
angle stiffeners spaced 60” apart. The objective is to investigate
duct‘s structural adequacy which is challenged by an exter
pressure of 5.26”w.g. (0.19 psi).
Several different methods are illustrated which empl
increasing levels of analytical complexity. Initially, ha
calculations are used to compute the elastic bending stresses in t
largest side panel and bordering angle stiffeners. This is follow
by the determination of panel plastic collapse using a closed for
solution that accounts for large deflection effects. Finally,
computerized non-linear finite element solution is obtained. For t
sample duct evaluated, the computer solution was the only metho
that was able to successfully confrm the structural adequacy of th
duct section.
2.0 DUCT CONFIGURATION
A sketch of the sample rectangular duct is shown in Fig.
The dimensions of the duct section including angle stiffeners a
given below
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Duct width = 116” or
q = Wt -W d = 0.169 psi (duct bottom panel)Duct height = 40”
Duct panel gauge = 14
(0.0747” nominal thickness)
Duct stiffeners= two structural angles, 2”x 3”x 3/16” (welded
back-to-back forming T-sections)
Stiffener spacing= 5’4’’ (typ.)
Duct panel material = ASTM A-570
(hot rolled carbon steel sheet)
Duct stiffener material = ASTM A36
Figure 1. Typical Rectangular HVAC Duct Section
3.0 CLOSED FORM ANALYSIS
3.1 Linear Elastic Analvsis
The duct panel bending stresses are computed using an analysis
model based on a rectangular plate with two long edges fixed (at
duct stiffeners) and two short edges simply supported (at duct
corners). The plate maximum bending stress due to uniform
pressure loading occurs at the center of the fixed edge and is
equivalent to’
j 9 x q x b 2= 67.7 ksi
t2
0-=
where
a = 116 in. (long side of the panel)
b = 60 in. (short side of the panel)
t = 0.0747 in. (panel nominal thickness)
p = 0.4972 (factor dependent on value of ah)
q = uniform load per unit area (pressure)
q = Wt +w d = 0.21 1 psi (duct top panel)
. (1)
Wt = 0.19 psi (vacuum pressure corresponding to tornado even
w d = 0.0211psi (equivalent panel pressure due to deadweight
Note -pressure units conversion: 1”w.g. = 0.0361 1 psi
The panel bending stress due to deadweight effects is includ
in the above formulation. The linear elastic computed stress of 6
ksi exceeds the ASME AG-1 Code (AA-4300) allowable value
25.3 ksi (ASTM A-570 @ 2.259) for this Level-D Service conditi
The stiffener bending stress for fixed end stiffeners
computed using the following equation per SMACNA’:
A4 q x s 2- 42.6 ksi
O b = -= ~-2 1 0 x 2
... 2)
where
4 = c X p X c = 12.7 Ib/in (load on stiffener)
S = 116 in . (panel width or length of stiffener)
L = 60 in. (stiffener spacing)
p = 0.211psi (top panel pressure including deadweight
C = 1.0for L / s I 2.0
2 = 0.401in3 two stiffeners, 3”x2”~3/16”welded
back-to-back)
effects)
The linear elastic computed stress of 42.6 ksi exceeds t
ASME AG-1 Code (AA4300) allowable value of 32.6 ksi (AST
A-36 @ 2.258) for thisLevel-D Service condition.
3.2 Lame Deflection, Collapse Load
This analysis method is based on several simplifjm
assumptions whereby a panel is analytically modeled as a f
rectangular plate with simply supported edges and a unifo
(pressure) load. The plate modulus of rupture is determin
analytically by computing (approximately) the ultimate lo
corresponding to the breaking point and then applying a safety fac
for evaluation purposes. Reference 5 (pg. 409) has solutions f
several plate configurations but suggests that this analytic
approach can be in error by as much as 30%. This computation
uncertainty can be accounted for in the evaluation safety factor.
ASME AG-1, Section AA-4300 allows structural evaluation
limit analysis and collapse load determination. For Level-D Serv
structural criteria,ASMEAG-1 refers to the ASME Code, Secti
III. ASME Code Sectiona ivision 1, Appendix F, Section
1331 requires that the actual component load shall not exceed 90
of the predicted collapse load using elastic analysis metho
Consequently, the evaluation safety factor should be def ied so as
account for both computational uncertainty and ASME Code marg
i.e., S.F.=0.70 x 0.90 = 0.63.
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The collapse uniform pressure load required to collapse the
plate is5:
pallownble= 0.63 x P,= 0.0773psi
where
wu= collapse load, lb.
O y = 25.0 ksi (ASTMA-570 Gr.A)
t = 0.0747 in. (panel thickness)
p = 6.11 (factor dependent on value of b/a)
a = 116 in. (width of the duct panel)
b = 60 in. (stiffener spacing)
0 . . (3)
. (4)
As indicated in Section 3.1 above, the actual total equivalent
pressure on the top panel is 0.211 psi (including deadweight
effects). This exceeds the ASME Code allowable Uniform pressure
load required to collapse the plate of 0.0773 psi.
The closed form solutions illustrated above are very
conservative in their determination of duct panel lateral pressure
load capability. Analysis using a computer is discussed in the
section that follows.
4.0 FINITE ELEMEN T ANALYSIS
4.1 Assumptions
The structural responses of adjacent duct sections are assumed
to be identical based on symmetry. As a result, only a single duct
section that i s bounded by a pair of stiffeners is analyzed (Fig. 2).
Symmetrical boundary conditions can be applied on the stiffeners at
the two edges of the duct section.
The material properties of the stiffeners (ASTM A36) are
assumed to be the same as those of the duct. The material stress
strain characteristics are discussed inAppendix A.
The steel angles are assumed to be Uniformly f i e d to the duct
plates (continuous integral welds).
4.2 MethodoloqvFinite-element static stress analyses were performed by using
the mAQU S6 computer program. Two finite element models were
generated using the AEiAQUS S4R 3-D shell elements - a full duct
section model and a one-eighth duct section model. The one-eighth
model takes advantage of the geometrical symmetry of the duct
section. Figures 2 and 3 show the full and the one-eighth models
togetha with their boundary conditions.
Three different types of analyses were performed for t
external pressure load, i.e., (1) linear elastic analysis; (2) elast
plastic, small deflection analysis; and (3) elastic-plastic, lar
deflection analysis.
For the case (2) and (3) nonlinear analyses, some difficulty w
encountered in obtaining solution convergence. This difficulty w
overcome by utilizing a combination of both direct and automa
time step selections to direct the iterative solutions. For examp
near the point of predicted structural instability, the revised Ri
method6was used to determine the corresponding applied load.
4.3 Summarv of Analytical Results
4.3.1. Full Duct Section Model
The results of the linear elastic analysis indicate tha t
maximum von Mises stress due to the tornado-induc
depressurization of 0.19 psi (5.26"w.g.) exceeds the AG-1 allowab
of 25.3 ksi. This represents a very conservative solution since,
addition to assuming a linear stress-strain relationship, the bendi
moment reductions due to large rotations are neglected. T
absence of membrane stiffness associated with large deflections al
contributes to analysis conservatism.
The results of the elastic-plastic, small-deflection analysis al
predicts that the maximum von Mises stress due to the tornad
induced depressurization of 0.19 psi (5.26"w.g.) exceeds the AG
criteria. This also represents a conservative solution because of t
limitations of small deflection theory.
The results of the elastic-plastic, large-deflection analy
indicates that the maximum stress due to the tomado-induc
depressurizationof
0.19 psi (5.26"w.g.) is actually below the yie
stress. Themaximumvon Mises stress was computed to be 14.4 k
which is acceptable per the AG-1 criteria. The maximum deflecti
was computed to be about 1.0". Figure 4 is an illustration of t
deformed shape (magnified) due to the applied external pressure.
x+y Figure 4. Deformed Shape of Duct Section Due to
External Pressure of 5 .26"w.g. (0.19 psi)
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Figure 5 shows the load-deflection curves of the duct section as
determined by the three types of analyses. The load is the applied
uniform external pressure, whereas the deflection corresponds to the
maximum duct deflection at the center of the bottom panel.
4.3.2 One-Eiahth Duct Section Model
The stress distribution predicted by the one-eighth model due
to the external pressure is the same as that for the full model. The
load-deflection curve of the one-eighth model is shown in Fig. 5.
When the external pressure is below 0.3 psi, (8.3”w.g.) the
load-deflection for the elastic-plastic, large-deflection analysis is
approximately the same as that of the full model. However, the
load-deflection curve deviates from that of the full-model with
increasing external pressure. The one-eighth model is not able to
predict the postbuckling mode identified by the full model, and thus,
can not predict the collapse load accurately.
5.0 DETERMINATION OF COLLAPSE LOAD
Tomado-induced depressurization is classified as a Level DService (Faulted) load (DPD) per ASME AG-13. Accordingly, the
acceptance criterion of collapse loads defined in the ASME Code,
Sectionm, ppendix F4 is applicable.
To account for post-buckling behavior in the determination of
duct collapse load, it is necessary to use the full model. The
resulting load-deflection curve is shown in Fig. 6. The allowable
collapse load according to the ASME Code is 13.2”w.g. (0.477 psi)
and is shown in the figure along with the expected external pressure
corresponding to the magnitude of tomado-induced depressurization,
5.26”w.g. (0.19 psi).
6.0 CONCLUSIONS
Conclusions from the present analysis are as follows:
1. The allowable collapse load was determined to be 13.2”w.g.
(0.477 psi) based on ASME Code, SectionID, ppendix F for
Level D (Faulted) Service4. The anticipated maximum external
pressure in this duct section during the tornado is 5.26”w.g.
(0.19 psi) which is well within the structural capability of this
duct.
2. The maximum von Mises stress in the duct is 14.4 ksi, which ismuch less than what is permitted by AG-1 for this material
(25.3ksi). The maximum deflection was computed to be about1O inch.
3. Based on several linear and nonlinear analyses performed, it
was concluded that the effects of large deflections and large
rotations are important considerations for this type of
evaluation.
7.0 REFERENCES
1. W A C Duct Construction Standards, Metal and Flexib
SMACNA, 1985.
2. Rectangular Industrial Duct Construction Standar
SMACNA, 1980.
3. ASME AG-1-1994, Code onNuclearAir and Gas Treatment
AG-la-1996 Addenda, ASME.
4. ASME Boiler& Pressure Vessel Code, SectionIII,Appendix
1995.
5. Roark, R. J. and Young, W.C., Formulas for Stress and Stra
SfhEd.,McGraw-Hill, 1975.
6 . ABAQUS/Standard User’s Manual, Vol. IIandJII, version 5.
7. Shigley, J. E. and Mitchell, L. D., Mechanical Engineer
Design, 4thEd., McGraw-Hill, 1983.
8. ASTM A570-75, Standard Specification or Hot-Rolled Carb
Steel Sheet, and Strip, Structural Quality,ASTM, 1975.
-APPENDIX A-
Material Properties of ASTM A570*
derived in the following.
The inelastic material properties needed for the analysis
The nominal yield stress and,Young’s modulus of ASTM
570 steel are:
ov = 25,OOOpsZ
E = 29 x 106psi
The nominal tensile strength and its corresponding nomin
strain are:
The rt--tionship
a,, = 45,OOOpsi
E,,, = 0.27in/ in
ztween nominal stress an^ -1ominal strain i
a, = E&, ...(Al)
Where 0, nominal stress
E, = nominal strainE = Young’s modulus
The true stress-true strain relation is7:
5t=KEF . A 2 )
where at = true stress
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K = strength coeficient
&t =truestrain
n = strain-hardeningexponent
The true stress can be expressed in terms of the nominal stress
and nonlinal strain as follows7:
0;= o,(l+ E , ) .. A3)
Furthermore, the true strain can be expressed in terms of
nominal strain as fo ~ows~:
ct = ln(l+ E , ) . A4)
Using Equations (Al) through (A4), the following values of the
strength coefficient, K , and strain-hardening exponent, n , are
determined from the values of ow On,,, , nd K :
K = 70,500psi
n = 0.1468
Consequently, the formula for the true stress - true strain of
A570 Grade A steel is:
ot = 70,500~~.1468 ...(A5)
Figure A1 shows the plot of the true stress - true strain curve
based on equation A5.
QJ 2 1 m o
b 1110
15WO
,001 0
. i _ _OW 00 1 00 2 00 3 OM 0 0 s 0 0 6 00 7 00 8 0 0 9 0.1 0.11 01 2 013 0.14
True Strain (YO)
Figure A . l True Stress vs . True Strain for ASTM A570,
Grade A Steel
calculated from the true total strains and true stresses by using t
followingequation:
...(A 6 )
The plastic strain data for the duct material (ASTM A-57
ASTM A-36) is tabulated in Table A1 below.
TableAI. Material Plastic Strain Data
True Plastic Strainrue Stress
25,000 0.0
I 32,390 I 0.00388 II 35,860 I 0.00876 II 39,700 I 0.01863 II 43,950 I 0.03848 II 46,650 I 0.05839 II 48,660 I 0.07832 II 50,280 1 0.09827 II 51,640 I 0.1182 II 52,830 I 0.1382 I
Acknowledgments
Contract DE-AC09-96SR18500.
This work was performed under U.S. Department of Ener
In performing elastic-plastic analysis, the ABAQUS computer
progran? requires material property data in the form of true stresses
and corresponding plastic strains. True plastic strains can be
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,
Figure 2. Full Model of 116"x 40" x 60" Rectangular Duct Section
Figure 3. One-Eighth Model of 116" x 40" x 60" Rectangular Duct Section
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0.8 ,Load Deflection Characteristics or 116"x 40" 60' Rectangular Duct
t lastoplastic,LargeDeflection, FullModel
Q Elastoplastic,J x g eDeflection, One-EighthModel
-& EWplastiO, Small Deflection
+t- LinearElastic
18-
16 --
14 --
12 --
c
5 io--
dt
Deflection (in)
Figure5. Load Deflection Curves for Various Analysis Cases
/
---C Load Deflection Curve- inear Load Deflection Line- ollapse Limit Line
----- Tornado Depressurization Load (5.26 in. w.g.)
I0 1 2 3 4 5 6 7 B 9 10
Deflection (in)
Figure 6. Determination of Allowable Collapse Load